K12MATH011: Algebra II

Unit 4: General Polynomial FunctionsHaving been introduced to first-power (linear) and second-power
(quadratic) forms, you are now ready to approach general polynomial
equations and functions. Polynomials are expressions that are made up of
sums and differences of terms, such that the coefficients are all real
numbers and the exponents on the variables are all whole numbers. This
unit begins with an introduction to polynomials in one variable,
stressing the degree (highest power) and what the degree may imply about
a polynomial function’s solutions. You will learn to perform basic
operations (adding, subtracting, multiplying, and dividing) on
polynomial expressions and how to apply these operations to the
composition of functions. You will learn how to graph polynomial
functions and end the unit with a fairly detailed method for finding all
the roots (solutions) of a polynomial.

Unit 4 Time Advisory
Completing this unit should take approximately 7 hours and 45 minutes.

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Subunit 4.1: 45 minutes

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Subunit 4.2: 2 hours and 30 minutes

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Subunit 4.3: 1 hour

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Subunit 4.4: 1 hour

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Subunit 4.5: 2 hours and 30 minutes

Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- Add, subtract, divide, and multiply polynomial functions.
- Find the compositions of functions.
- Use the composition of two functions to determine whether they are
inverses.
- Determine the basic graph of a polynomial function.
- Find the roots of a polynomial function using Descartes’s method.

4.1 Introduction to PolynomialsThe most common of all functions that are used in the scientific and
business communities today are polynomials. This is because they are
easy to use and have centuries of study upon them. Polynomials are
single terms or sums and differences of terms, such that the powers on
all possible variables are whole numbers. That is, if there is a
variable (and there doesn’t have to be), the variable must have a whole
number power or else it’s not a polynomial.

4.2 Operations with PolynomialsSince polynomials may represent real numbers, polynomial functions may
be added, subtracted, multiplied, and divided. Polynomial functions play
an important role in business and the sciences, as many processes can be
precisely modeled using polynomial functions. For this reason, we need
to know as much about working with polynomial equations as possible.
Also, techniques learned now will be useful later, as you will see these
techniques again and again.

Instructions: This video discusses the general process of dividing
polynomials.

Watching the video and writing notes should take approximately 30
minutes.

Standards Addressed (Common Core):

- [CCSS.Math.Content.HSA-APR.B.2](http://www.corestandards.org/Math/Content/HSA/APR/B/2)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 United States
License](http://creativecommons.org/licenses/by-nc-sa/3.0/us/). It
is attributed to Khan Academy.

Instructions: This video discusses the process of synthetic
division. Note that unlike general polynomial division, this only
works for dividing a polynomial by a binomial. This is a fast way to
determine if a particular value is a root of the polynomial (the
remainder is zero for synthetic division, in that case).

Watching the video and writing notes should take approximately 15
minutes.

4.4 Polynomial Equations and FunctionsPolynomials form smooth curves with no breaks or skips. Their graphs
have no breaks in them and their extremes are dominated by the highest
power term. For example, the function ƒ(x) = 2x3 +
5x2 – 7x + 1 behaves just like ƒ(x) = 2x3 as
the values of x become more and more positive or negative. The following
material will make it easy to understand and graph these equations and
functions.

Did I Get This? Activity: Khan Academy’s “Views of a Function”
Link: Khan Academy’s “Views of a
Function”
(HTML)

Instructions: This is a self-quiz on general functions, including
polynomials (which include linear and quadratic functions). Note
that there are hints available if needed.

Completing the practice quiz and writing notes should take
approximately 1 hour.

4.5 Finding Roots (Solutions) to Polynomial FunctionsFinding the roots of any polynomial is possible, but it requires time
to master. In general, great minds came up with methods for solving
polynomials nearly 300 years ago. The methods are still useful today and
can be translated to computer programming.

Instructions: Review the examples, practice problems, and videos on
the accumulated methods for finding all roots (solutions) for any
given polynomial. The material article covers the rational root
theorem as well as the use of synthetic division to quickly find all
real roots. Note: Once all real roots are exhausted, what remains,
in pairs, will be irrational roots. Methods for finding those are
demonstrated as well. This is an accumulation of all techniques
learned to this point. Now, consider conjugate pairs of complex
numbers (like 2 -3i and 2 + 3i or 7 + 5i and 7 – 5i). Experiment
with pairs of conjugate complex roots and single or odd numbers of
complex roots. Write a one-page essay explaining why any complex
roots of a polynomial with real coefficients must be in conjugate
pairs.

Completing this activity should take approximately 1 hour and 30
minutes.

Instructions: This is an assessment on understanding polynomials.
Work all 26 problems. Answers are available for viewing by a button
on each problem. There is also a link for a printed version at the
bottom left of the page. Note: New versions of the 26 problems can
be created by accessing the link in the upper right corner of the
page. You may take a nearly unlimited number of attempts with
different problems.